In [1]}, there is also an analytic version of cohomological Donaldson-Thomas theory, for analytic oriented d-critical analytic spaces, which are locally the critical locus of an holomorphic function. These constructions can also be extended to d-critical oriented analytic stacks using an analytic version of the constructions of [2]}. Unfortunately we are missing crucial results on \(\mathbb {C}^\ast \) -action on analytic spaces or stacks. When an analytic space (resp. stack) with \(\mathbb {C}^\ast \) -action has a \(\mathbb {C}^\ast \) -equivariant analytic (resp. smooth) cover by an affine analytic space, we show that the fixed and attracting/repelling variety are algebraic spaces (resp. stack), hence the hyperbolic localization functors, and the morphism \(S_X\) , are defined. However, we do not know under which generality such a \(\mathbb {C}^\ast \) -equivariant cover exists, and, according to David Rydh (private communication), the main techniques of [3]} does not apply directly to the analytic setting. Moreover, we need an analytic version of [4]} saying that \(S_X\)  is an isomorphism when applied to \(\mathbb {C}^\ast \) -equivariant complexes. Furthermore, according to [1]}, the six-functor formalism is not fully defined for mixed Hodge modules in the analytic case, hence it could be difficult to lift a localization result from the level of perverse sheaves to the level of mixed Hodge modules. Once such results are available, the whole results of this article should extend directly to the analytic case.
